Uninformed search strategies A search strategy is defined by picking the order of node expansion Uninformed search strategies use only the information available in the problem definition Breadth-first search Depth-first search Iterative deepening search Uniform-cost search

Example state space graph for a tiny search problem Breadth-first search Expand shallowest unexpanded node Implementation: frontier is a FIFO queue Example state space graph for a tiny search problem Example from P. Abbeel and D. Klein

Breadth-first search Expansion order: (S,d,e,p,b,c,e,h,r,q,a,a, h,r,p,q,f,p,q,f,q,c,G)

Depth-first search Expand deepest unexpanded node Implementation: frontier is a LIFO queue

Depth-first search Expansion order: (d,b,a,c,a,e,h,p,q,q, r,f,c,a,G)

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Analysis of search strategies Strategies are evaluated along the following criteria: Completeness: does it always find a solution if one exists? Optimality: does it always find a least-cost solution? Time complexity: number of nodes generated Space complexity: maximum number of nodes in memory Time and space complexity are measured in terms of b: maximum branching factor of the search tree d: depth of the optimal solution m: maximum length of any path in the state space (may be infinite)

Properties of breadth-first search Complete? Yes (if branching factor b is finite) Optimal? Yes – if cost = 1 per step Time? Number of nodes in a b-ary tree of depth d: O(bd) (d is the depth of the optimal solution) Space? O(bd) Space is the bigger problem (more than time)

Properties of depth-first search Complete? Fails in infinite-depth spaces, spaces with loops Modify to avoid repeated states along path  complete in finite spaces Optimal? No – returns the first solution it finds Time? Could be the time to reach a solution at maximum depth m: O(bm) Terrible if m is much larger than d But if there are lots of solutions, may be much faster than BFS Space? O(bm), i.e., linear space!

Iterative deepening search Use DFS as a subroutine Check the root Do a DFS searching for a path of length 1 If there is no path of length 1, do a DFS searching for a path of length 2 If there is no path of length 2, do a DFS searching for a path of length 3…

Iterative deepening search

Iterative deepening search

Iterative deepening search

Iterative deepening search

Properties of iterative deepening search Complete? Yes Optimal? Yes, if step cost = 1 Time? (d+1)b0 + d b1 + (d-1)b2 + … + bd = O(bd) Space? O(bd)

Search with varying step costs BFS finds the path with the fewest steps, but does not always find the cheapest path

Uniform-cost search For each frontier node, save the total cost of the path from the initial state to that node Expand the frontier node with the lowest path cost Implementation: frontier is a priority queue ordered by path cost Equivalent to breadth-first if step costs all equal Equivalent to Dijkstra’s algorithm in general

Uniform-cost search example

Uniform-cost search example Expansion order: (S,p,d,b,e,a,r,f,e,G)

Another example of uniform-cost search Source: Wikipedia

Properties of uniform-cost search Complete? Yes, if step cost is greater than some positive constant ε (we don’t want infinite sequences of steps that have a finite total cost) Optimal? Yes

Optimality of uniform-cost search Graph separation property: every path from the initial state to an unexplored state has to pass through a state on the frontier Proved inductively Optimality of UCS: proof by contradiction Suppose UCS terminates at goal state n with path cost g(n) = C but there exists another goal state n’ with g(n’) < C Then there must exist a node n” on the frontier that is on the optimal path to n’ But because g(n”) ≤ g(n’) < g(n), n” should have been expanded first!

Properties of uniform-cost search Complete? Yes, if step cost is greater than some positive constant ε (we don’t want infinite sequences of steps that have a finite total cost) Optimal? Yes – nodes expanded in increasing order of path cost Time? Number of nodes with path cost ≤ cost of optimal solution (C*), O(bC*/ ε) This can be greater than O(bd): the search can explore long paths consisting of small steps before exploring shorter paths consisting of larger steps Space? O(bC*/ ε)